That would really restrict what you could buy. I picked up the Amazon Fire TV HDTV and it was on a black friday delivered to my door at about 312USD total. 50 inches but the wired headset seems to drop out so that's a bug. Too bad because so much else works so well. I eventually added a sound bar by Toshiba since Toshiba made the HDTV and to my surprise ARC works correctly. Even controls my BD player.

The Roku just plugs into any HDMI port so that's something all the sets have.

As to the battery time, no HDTV I know comes with wireless headsets so the time on battery is something you check into on its own.

-> So what next? Make a list of 400 to 500 buck models then check to see if they have the headphone jack and mention Bluetooth headphone support. You won't find many.

If you don't want to do that work, just go to Bestbuy and tell them your requirements. Get the model numbers and check again they have the required features.

Sadly I have yet to find a HDTV finder/shopper that does this for us.